Probability distribution of residence-times of grains in sandpile models

TL;DR

This paper analyzes the probability distribution of grain residence times in sandpile models, linking it to diffusion processes, and provides exact solutions for specific cases along with general results for large systems.

Contribution

It introduces a novel connection between residence-time distributions in sandpiles and diffusion survival probabilities, providing exact solutions for 1D cases and general asymptotic results.

Findings

Exact scaling function for 1D sandpile with end-only grain addition.

Exponential tail of residence time distribution for uniform addition in large systems.

Finite-size corrections to the residence-time distribution.

Abstract

We show that the probability distribution of the residence-times of sand grains in sandpile models, in the scaling limit, can be expressed in terms of the survival probability of a single diffusing particle in a medium with absorbing boundaries and space-dependent jump rates. The scaling function for the probability distribution of residence times is non-universal, and depends on the probability distribution according to which grains are added at different sites. We determine this function exactly for the 1-dimensional sandpile when grains are added randomly only at the ends. For sandpiles with grains are added everywhere with equal probability, in any dimension and of arbitrary shape, we prove that, in the scaling limit, the probability that the residence time greater than t is exp(-t/M), where M is the average mass of the pile in the steady state. We also study finite-size corrections to this function.